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Joaquim Martin

Publications and source records attributed to Joaquim Martin.

At least 19 recordsLinked to original sources
arXiv2026

Oscillation Classes: An Interpolation Approach

Let \(φ\) be an admissible concave function on \((0,1)\) and let \(X\) be a rearrangement-invariant space. We study the classes determined by the oscillation functional \[ \mathcal N_{φ,X}(f) = \left\| \frac{f^{**}-f^*}φ \right\|_X+\|f\|_1. \] We develop an interpolation method, based on the Aronszajn--Gagliardo extremal construction, which allows the normability problem and the determination of the optimal rearrangement-invariant Banach exterior to be treated in a unified way. A recovery principle shows that the oscillation construction reflects the inclusion order of the underlying rearrangement-invariant spaces. This makes it possible to transfer the corresponding Aronszajn--Gagliardo extremal structure to the oscillation classes. In particular, the upper extremal generates the least rearrangement-invariant Banach space containing the class, while normability occurs precisely when the lower and upper extremals collapse. At the critical fundamental scale, normability is rigid and forces the underlying space to be the corresponding Lorentz endpoint. Applications to Lorentz, limiting Lorentz, and Orlicz scales illustrate the scope of the method, including limiting normable examples for which the classical Copson absorption mechanism fails.

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arXiv2026

Oscillation Functionals and Embeddings in Rearrangement-Invariant Spaces

We study embeddings associated with oscillation functionals in rearrangement-invariant spaces. More precisely, given a positive function \(ψ\), we analyze how the interaction between the geometry of the underlying space and the growth of \(ψ\) determines the behaviour of these embeddings, leading to a natural classification into subcritical, supercritical and critical regimes. We prove that in the critical regime logarithmic refinements of Hansson type appear, governed by a deviation function associated with the quotient \(ψ/φ_X\), where \(φ_X\) is the fundamental function of the underlying space. This leads to explicit Hansson-type targets and, in the bounded case of the deviation function, to Trudinger-type consequences. The results recover and extend several classical endpoint embeddings.

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arXiv2025

Generalised Hajłasz-Besov spaces on $RD$-spaces

An $RD$ space is a doubling measure metric space $Ω$ with the additional property that it has a reverse doubling property. In this paper we introduce a new class of Hajłasz-Besov spaces on $Ω$ and extend several results from classical theory, such as embeddings and Sobolev-type embeddings.

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arXiv2022

Non-collapsing condition and Sobolev embeddings for Hajłasz-Besov spaces

In this paper we will focus on understanding the relation between Sobolev embedding theorems for Hajłasz-Besov spaces defined on a doubling metric measure space $(Ω,d,μ)$ and the non-collapsing condition of the measure, i.e. \[ \inf_{x\inΩ}μ(B(x,1))>0. \] We will also obtain embedding results for Hajłasz-Besov spaces whose modulus of smoothness is generated by a rearrangement invariant quasi-norm.

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arXiv2016

Isoperimetric weights and generalized uncertainty inequalities in metric measure spaces

We extend the recent $L^{1}$ uncertainty inequalities obtained by Dall'ara-Trevisan to the metric setting. For this purpose we introduce a new class of weights, named *isoperimetric weights*, for which the growth of the measure of their level sets $μ(\{w\leq r\})$ can be controlled by $rI(r),$ where $I$ is the isoperimetric profile of the ambient metric space. We use isoperimetric weights, new *localized Poincaré inequalities*, and interpolation, to prove $L^{p},1\leq p<\infty,$ uncertainty inequalities on metric measure spaces. We give an alternate characterization of the class of isoperimetric weights in terms of Marcinkiewicz spaces, which combined with the sharp Sobolev inequalities we had obtained in an earlier paper, and interpolation of weighted norm inequalities, give new uncertainty inequalities in the context of rearrangement invariant spaces.

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arXiv2014

Towards a unified theory of Sobolev inequalities

We discuss our work on pointwise inequalities for the gradient which are connected with the isoperimetric profile associated to a given geometry. We show how they can be used to unify certain aspects of the theory of Sobolev inequalities. In particular, we discuss our recent papers on fractional order inequalities, Coulhon type inequalities, transference and dimensionless inequalities and our forthcoming work on sharp higher order Sobolev inequalities that can be obtained by iteration.

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arXiv2014

Fractional Sobolev Inequalities: Symmetrization, Isoperimetry and Interpolation

We obtain new oscillation inequalities in metric spaces in terms of the Peetre $K-$functional and the isoperimetric profile. Applications provided include a detailed study of Fractional Sobolev inequalities and the Morrey-Sobolev embedding theorems in different contexts. In particular we include a detailed study of Gaussian measures as well as probablity measures between Gaussian and exponential. We show a kind of reverse Polya-Szego principle that allows us to obtain continuity as a self improvement from boundedness, using symetrization inequalities. Our methods also allow for precise estimates of growth envelopes of generalized Sobolev and Besov spaces on metric spaces. We also consider embeddings into $BMO$ and their connection to Sobolev embeddings.

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arXiv2013

Integral isoperimetric transference and dimensionless Sobolev inequalities

We introduce the concept of Gaussian integral isoperimetric transference and show how it can be applied to obtain a new class of sharp Sobolev-Poincaré inequalities with constants independent of the dimension. In the special case of $L^{q}$ spaces on the unit $n-$dimensional cube our results extend the recent inequalities that were obtained in \cite{FKS} using extrapolation.

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arXiv2013

A Note on Coulhon type inequalities

T. Coulhon introduced an interesting reformulation of the usual Sobolev inequalities. We characterize Coulhon type inequalities in terms of rearrangement inequalities.

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arXiv2009

Isoperimetry and Symmetrization for Sobolev spaces on metric spaces

Using isoperimetry we obtain new symmetrization inequalities that allow us to provide a unified framework to study Sobolev inequalities in metric spaces. The applications include concentration inequalities, as well as metric versions of the Pó% lya-Szegö and Faber-Krahn principles.

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arXiv2008

Isoperimetry and symmetrization for logarithmic Sobolev inequalities

Using isoperimetry and symmetrization we provide a unified framework to study the classical and logarithmic Sobolev inequalities. In particular, we obtain new Gaussian symmetrization inequalities and connect them with logarithmic Sobolev inequalities. Our methods are very general and can be easily adapted to more general contexts.

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